## Introduction to Fourier Transform

The Fourier Transform is a powerful mathematical tool used to analyze and represent functions in terms of their frequency components. It transforms a time-domain signal into its frequency-domain representation, providing insights into the signal’s frequency content.

The development of the Fourier Transform is a fascinating journey through the history of mathematics and physics, beginning in the 18th century and evolving through the contributions of many brilliant minds. This section delves into the historical context and milestones that led to the formulation and widespread application of the Fourier Transform.

#### Joseph Fourier and the Birth of Fourier Analysis

The Fourier Transform is named after Jean-Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist. Fourier’s pioneering work in the early 19th century laid the foundation for what we now call Fourier analysis.

##### Key Contributions:
• Heat Conduction: Fourier’s interest in the mathematical description of heat conduction led to his groundbreaking work. He proposed that any periodic function could be expressed as a sum of sines and cosines, each with specific amplitudes and frequencies. This idea was initially met with skepticism, but it eventually gained acceptance due to its utility and accuracy.
• Fourier Series: Fourier introduced the concept of Fourier series in his seminal work “Théorie analytique de la chaleur” (The Analytical Theory of Heat) published in 1822. This work demonstrated how to decompose periodic functions into infinite sums of sine and cosine functions, known as Fourier series.

$f(t) = a_0 + \sum_{n=1}^{\infty}(a_n cos(n\omega_{0}t) + b_n sin(n\omega_{0}t))$

#### Mathematical Formalization and Extensions

While Fourier’s initial work focused on periodic functions, subsequent mathematicians extended and formalized his ideas, leading to the development of the Fourier Transform for non-periodic functions.

##### Key Figures and Contributions:
• Dirichlet and Riemann: Johann Peter Gustav Lejeune Dirichlet (1805-1859) and Bernhard Riemann (1826-1866) made significant contributions to the formalization of Fourier series, providing rigorous mathematical foundations and conditions for their convergence.
• Henri Lebesgue: In the early 20th century, Henri Lebesgue (1875-1941) further generalized Fourier’s work by developing Lebesgue integration, which provided a more robust framework for dealing with functions that could be decomposed into Fourier series.
• Marcel Riesz: Marcel Riesz (1886-1969) made important contributions to the theory of the Fourier Transform, particularly in the context of functional analysis and the theory of distributions.

#### Development of the Fourier Transform

The transition from Fourier series to the Fourier Transform involved extending the concept to non-periodic functions, leading to the integral form of the Fourier Transform that we use today.

##### Key Developments:
• Non-Periodic Functions: Mathematicians realized that Fourier’s idea of decomposing functions into sines and cosines could be extended to non-periodic functions by considering the limit as the period approaches infinity. This led to the formulation of the Fourier Transform as an integral.
• Heaviside and Signal Processing: In the late 19th and early 20th centuries, Oliver Heaviside (1850-1925) and other engineers began using Fourier methods in electrical engineering and signal processing. Heaviside’s operational calculus and work on the Heaviside step function further popularized the use of Fourier analysis in engineering.

## Definition of Fourier Transform

##### Fourier Transform

Given a function $$f(t)$$, the Fourier Transform $$F(\omega)$$ is defined as:

$F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} \, dt.$

Here, $$\omega$$ represents the angular frequency, and $$i$$ is the imaginary unit.

##### Inverse Fourier Transform

The inverse Fourier Transform allows us to reconstruct the original time-domain function from its frequency-domain representation. It is defined as:

$f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i \omega t} \, d\omega.$

## Properties of Fourier Transform

The Fourier Transform has several important properties that make it useful for various applications in engineering, physics, and signal processing. Here are some key properties along with short proofs.

##### 1. Linearity

If $$f(t)$$ and $$g(t)$$ are two functions, and $$a$$ and $$b$$ are constants, then:

$\mathcal{F} { a f(t) + b g(t) } = a F(\omega) + b G(\omega).$

##### Proof:

\begin{aligned} \mathcal{F} { a f(t) + b g(t) } &= \int_{-\infty}^{\infty} [a f(t) + b g(t)] e^{-i \omega t} \, dt \\ &= a \int_{-\infty}^{\infty} f(t) e^{-i \omega t} \, dt + b \int_{-\infty}^{\infty} g(t) e^{-i \omega t} \, dt \\ &= a F(\omega) + b G(\omega). \end{aligned}

##### 2. Time Shifting

If $$f(t)$$ is shifted by $$t_0$$​, then:

$\mathcal{F} { f(t – t_0) } = e^{-i \omega t_0} F(\omega).$

##### Proof:

\begin{aligned} \mathcal{F} { f(t – t_0) } &= \int_{-\infty}^{\infty} f(t – t_0) e^{-i \omega t} \, dt \\ &= \int_{-\infty}^{\infty} f(u) e^{-i \omega (u + t_0)} \, du \quad (\text{Let } u = t – t_0) \\ &= e^{-i \omega t_0} \int_{-\infty}^{\infty} f(u) e^{-i \omega u} \, du \\ &= e^{-i \omega t_0} F(\omega). \end{aligned}

##### 3. Frequency Shifting

If $$f(t)$$ is multiplied by $$e^{i \omega_0 t}$$, then:

##### Proof:

\begin{aligned} \mathcal{F} { f(t) e^{i \omega_0 t} } &= \int_{-\infty}^{\infty} f(t) e^{i \omega_0 t} e^{-i \omega t} \, dt \\ &= \int_{-\infty}^{\infty} f(t) e^{-i (\omega – \omega_0) t} \, dt \\ &= F(\omega – \omega_0). \end{aligned}

##### 4. Convolution Theorem

The Fourier Transform of the convolution of two functions $$f(t)$$ and $$g(t)$$ is the product of their individual Fourier Transforms:

$\mathcal{F} { f(t) * g(t) } = F(\omega) G(\omega).$

##### Proof:

\begin{aligned} \mathcal{F} { f(t) * g(t) } &= \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty} f(\tau) g(t – \tau) \, d\tau \right) e^{-i \omega t} \, dt \\ &= \int_{-\infty}^{\infty} f(\tau) \left( \int_{-\infty}^{\infty} g(t – \tau) e^{-i \omega t} \, dt \right) d\tau \\ &= \int_{-\infty}^{\infty} f(\tau) e^{-i \omega \tau} \, d\tau \int_{-\infty}^{\infty} g(u) e^{-i \omega u} \, du \\ &= F(\omega) G(\omega). \end{aligned}

##### 5. Parseval’s Theorem

The total energy of a signal in the time domain is equal to the total energy in the frequency domain:

$\int_{-\infty}^{\infty} |f(t)|^2 \, dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |F(\omega)|^2 \, d\omega.$

##### Proof:

\begin{aligned} \int_{-\infty}^{\infty} |f(t)|^2 \, dt &= \int_{-\infty}^{\infty} f(t) \overline{f(t)} \, dt \\ &= \int_{-\infty}^{\infty} \left( \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i \omega t} \, d\omega \right) \left( \frac{1}{2\pi} \int_{-\infty}^{\infty} \overline{F(\omega’)} e^{-i \omega’ t} \, d\omega’ \right) dt \\ &= \frac{1}{2\pi} \int_{-\infty}^{\infty} |F(\omega)|^2 \, d\omega. \end{aligned}

## Connection with Dirac Delta Function

The Dirac delta function, $$\delta(t)$$, is a generalized function with the following properties:

• $$\delta(t) = 0$$ for $$t\neq0$$
• $$\int_{-\infty}^{\infty}\delta(t)dt = 1$$.

It can be thought of as an infinitely narrow and infinitely tall spike at $$t\ = 0$$ with an area of $$1$$ under the curve.

##### Fourier Transform of Dirac Delta Function

The Fourier Transform of the Dirac delta function is a constant function:

$\mathcal{F} { \delta(t) } = 1.$

##### Proof:

\begin{aligned} \mathcal{F} { \delta(t) } &= \int_{-\infty}^{\infty} \delta(t) e^{-i \omega t} \, dt \\ &= e^{-i \omega \cdot 0} \\ &= 1. \end{aligned}

##### Inverse Fourier Transform of Dirac Delta Function

The inverse Fourier Transform of $$1$$ is the Dirac delta function:

$\mathcal{F}^{-1} { 1 } = \delta(t).$

##### Proof:

\begin{aligned} \mathcal{F}^{-1} { 1 } &= \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{i \omega t} \, d\omega \\ &= \delta(t). \end{aligned}

The Fourier Transform is a fundamental tool in signal processing and many other fields. It allows us to decompose signals into their constituent frequencies and analyze them in the frequency domain. The inverse Fourier Transform enables the reconstruction of the original signal. The properties of the Fourier Transform, such as linearity, time shifting, frequency shifting, convolution theorem, and Parseval’s theorem, provide powerful methods for signal analysis and manipulation. The connection with the Dirac delta function highlights the utility of the Fourier Transform in dealing with generalized functions and impulses.

By understanding and applying these concepts, we can gain deeper insights into the behavior of signals and systems in both the time and frequency domains.

## Fourier Transform in Modern Mathematics and Science

Throughout the 20th century, the Fourier Transform became an indispensable tool in various fields of science and engineering, with numerous applications and extensions.

##### Key Applications and Extensions:
• Quantum Mechanics: The Fourier Transform plays a crucial role in quantum mechanics, particularly in the formulation of wave functions and the analysis of quantum states.
• Signal Processing: In signal processing, the Fourier Transform is used for analyzing and filtering signals, leading to developments in telecommunications, audio engineering, and image processing.
• Medical Imaging: Techniques such as MRI (Magnetic Resonance Imaging) and CT (Computed Tomography) scans rely on the Fourier Transform to reconstruct images from raw data.
• Mathematical Analysis: The development of the Fast Fourier Transform (FFT) algorithm by James Cooley and John Tukey in 1965 revolutionized the practical application of Fourier analysis by making it computationally efficient.

## Conclusion

The history of the Fourier Transform is a testament to the power of mathematical ideas to transcend their original context and find applications in a wide array of scientific and engineering disciplines. From Joseph Fourier’s initial insights into heat conduction to the modern-day applications in signal processing and quantum mechanics, the Fourier Transform has become a cornerstone of mathematical analysis and a vital tool for understanding and manipulating complex functions and signals. Its continued relevance and adaptability underscore its foundational importance in both theoretical and applied mathematics.